reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 for i be Nat st 1 <= i & i < len Gauge(C,n) holds not
  Gauge(C,n)*(i,width Gauge(C,n)) in L~Lower_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  set wi = width Gauge(C,n);
  let i be Nat such that
A1: 1 <= i & i < len Gauge(C,n) and
A2: Gauge(C,n)*(i,wi) in L~Lower_Seq(C,n);
  set Gi1 = Gauge(C,n)*(i,wi);
  consider ii be Nat such that
A3: 1 <= ii and
A4: ii+1 <= len Lower_Seq(C,n) and
A5: Gi1 in LSeg(Lower_Seq(C,n),ii) by A2,SPPOL_2:13;
A6: LSeg(Lower_Seq(C,n),ii) = LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)
  ) by A3,A4,TOPREAL1:def 3;
  ii+1 >= 1 by NAT_1:11;
  then
A7: ii+1 in dom Lower_Seq(C,n) by A4,FINSEQ_3:25;
  len Gauge(C,n) >= 4 by JORDAN8:10;
  then len Gauge(C,n) = width Gauge(C,n) & len Gauge(C,n) > 1 by JORDAN8:def 1
,XXREAL_0:2;
  then
A8: [i,wi] in Indices Gauge(C,n) by A1,MATRIX_0:30;
  ii < len Lower_Seq(C,n) by A4,NAT_1:13;
  then
A9: ii in dom Lower_Seq(C,n) by A3,FINSEQ_3:25;
A10: not Gi1 in rng Lower_Seq(C,n) by A1,Th43;
  Lower_Seq(C,n) is_sequence_on Gauge(C,n) by Th5;
  then consider i1,j1,i2,j2 being Nat such that
A11: [i1,j1] in Indices Gauge(C,n) and
A12: Lower_Seq(C,n)/.ii = Gauge(C,n)*(i1,j1) and
A13: [i2,j2] in Indices Gauge(C,n) and
A14: Lower_Seq(C,n)/.(ii+1) = Gauge(C,n)*(i2,j2) and
A15: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
  or i1 = i2 & j1 = j2+1 by A3,A4,JORDAN8:3;
A16: 1 <= i1 by A11,MATRIX_0:32;
A17: j2 <= width Gauge(C,n) by A13,MATRIX_0:32;
A18: 1 <= j1 by A11,MATRIX_0:32;
A19: i1 <= len Gauge(C,n) by A11,MATRIX_0:32;
A20: 1 <= j2 by A13,MATRIX_0:32;
A21: i2 <= len Gauge(C,n) by A13,MATRIX_0:32;
A22: 1 <= i2 by A13,MATRIX_0:32;
A23: j1 <= width Gauge(C,n) by A11,MATRIX_0:32;
  per cases by A15;
  suppose
A24: i1 = i2 & j2+1 = j1;
    then j1 >= j2 by NAT_1:11;
    then Gauge(C,n)*(i1,j1)`2 >= Gauge(C,n)*(i2,j2)`2 by A16,A19,A23,A20,A24,
SPRECT_3:12;
    then
A25: Gauge(C,n)*(i1,j1)`2 >= Gi1`2 by A5,A6,A12,A14,TOPREAL1:4;
    Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A16,A19,A18,A23,A24,
GOBOARD5:2
      .= Gauge(C,n)*(i2,j2)`1 by A22,A21,A20,A17,GOBOARD5:2;
    then
    LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is vertical by A12,A14,
SPPOL_1:16;
    then Gi1`1 = Gauge(C,n)*(i1,j1)`1 by A5,A6,A12,SPPOL_1:41;
    then
A26: i1 = i by A11,A8,Th7;
    then Gi1`2 >= Gauge(C,n)*(i1,j1)`2 by A16,A19,A18,A23,SPRECT_3:12;
    then j1 = wi by A11,A8,A25,Th6,XXREAL_0:1;
    hence contradiction by A12,A9,A10,A26,PARTFUN2:2;
  end;
  suppose
A27: i2+1 = i1 & j1 = j2;
    then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A16,A19,A18,A23,
GOBOARD5:1
      .= Gauge(C,n)*(i2,j2)`2 by A22,A21,A20,A17,GOBOARD5:1;
    then LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is horizontal by A12
,A14,SPPOL_1:15;
    then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A5,A6,A12,SPPOL_1:40;
    then
A28: j1 = wi by A11,A8,Th6;
    i2 < len Gauge(C,n) by A19,A27,NAT_1:13;
    then not Lower_Seq(C,n)/.(ii+1) in rng Lower_Seq(C,n) by A14,A22,A27,A28
,Th43;
    hence contradiction by A7,PARTFUN2:2;
  end;
  suppose
A29: i2 = i1+1 & j1 = j2;
    then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A16,A19,A18,A23,
GOBOARD5:1
      .= Gauge(C,n)*(i2,j2)`2 by A22,A21,A20,A17,GOBOARD5:1;
    then LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is horizontal by A12
,A14,SPPOL_1:15;
    then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A5,A6,A12,SPPOL_1:40;
    then
A30: j1 = wi by A11,A8,Th6;
    i1 < len Gauge(C,n) by A21,A29,NAT_1:13;
    then not Lower_Seq(C,n)/.ii in rng Lower_Seq(C,n) by A12,A16,A30,Th43;
    hence contradiction by A9,PARTFUN2:2;
  end;
  suppose
A31: i1 = i2 & j2 = j1+1;
    then j2 >= j1 by NAT_1:11;
    then Gauge(C,n)*(i2,j2)`2 >= Gauge(C,n)*(i1,j1)`2 by A16,A19,A18,A17,A31,
SPRECT_3:12;
    then
A32: Gauge(C,n)*(i2,j2)`2 >= Gi1`2 by A5,A6,A12,A14,TOPREAL1:4;
    Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A16,A19,A18,A23,A31,
GOBOARD5:2
      .= Gauge(C,n)*(i2,j2)`1 by A22,A21,A20,A17,GOBOARD5:2;
    then
    LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is vertical by A12,A14,
SPPOL_1:16;
    then Gi1`1 = Gauge(C,n)*(i1,j1)`1 by A5,A6,A12,SPPOL_1:41;
    then
A33: i1 = i by A11,A8,Th7;
    then Gi1`2 >= Gauge(C,n)*(i2,j2)`2 by A22,A21,A20,A17,A31,SPRECT_3:12;
    then j2 = wi by A13,A8,A32,Th6,XXREAL_0:1;
    hence contradiction by A14,A7,A10,A31,A33,PARTFUN2:2;
  end;
end;
